Difference between revisions of "2020 AIME I Problems"
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==Problem 1== | ==Problem 1== | ||
− | + | In <math>\triangle ABC</math> with <math>AB=AC,</math> point <math>D</math> lies strictly between <math>A</math> and <math>C</math> on side <math>\overline{AC},</math> and point <math>E</math> lies strictly between <math>A</math> and <math>B</math> on side <math>\overline{AB}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | |
[[2020 AIME I Problems/Problem 1 | Solution]] | [[2020 AIME I Problems/Problem 1 | Solution]] | ||
==Problem 2== | ==Problem 2== | ||
− | + | There is a unique positive real number <math>x</math> such that the three numbers <math>\log_8(2x),\log_4x,</math> and <math>\log_2x,</math> in that order, form a geometric progression with positive common ratio. The number <math>x</math> can be written as <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | |
[[2020 AIME I Problems/Problem 2 | Solution]] | [[2020 AIME I Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | + | A positive integer <math>N</math> has base-eleven representation <math>\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}</math> and base-eight representation <math>\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},</math> where <math>a,b,</math> and <math>c</math> represent (not necessarily distinct) digits. Find the least such <math>N</math> expressed in base ten. | |
[[2020 AIME I Problems/Problem 3 | Solution]] | [[2020 AIME I Problems/Problem 3 | Solution]] | ||
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==Problem 13== | ==Problem 13== | ||
− | Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p | + | Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p</math>. |
[[2020 AIME I Problems/Problem 13 | Solution]] | [[2020 AIME I Problems/Problem 13 | Solution]] |
Latest revision as of 05:07, 8 March 2021
2020 AIME I (Answer Key) | AoPS Contest Collections | ||
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Contents
Problem 1
In with point lies strictly between and on side and point lies strictly between and on side such that The degree measure of is where and are relatively prime positive integers. Find
Problem 2
There is a unique positive real number such that the three numbers and in that order, form a geometric progression with positive common ratio. The number can be written as where and are relatively prime positive integers. Find
Problem 3
A positive integer has base-eleven representation and base-eight representation where and represent (not necessarily distinct) digits. Find the least such expressed in base ten.
Problem 4
Let be the set of positive integers with the property that the last four digits of are and when the last four digits are removed, the result is a divisor of For example, is in because is a divisor of Find the sum of all the digits of all the numbers in For example, the number contributes to this total.
Problem 5
Six cards numbered through are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.
Problem 6
A flat board has a circular hole with radius and a circular hole with radius such that the distance between the centers of the two holes is . Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is , where and are relatively prime positive integers. Find .
Problem 7
A club consisting of men and women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as member or as many as members. Let be the number of such committees that can be formed. Find the sum of the prime numbers that divide
Problem 8
A bug walks all day and sleeps all night. On the first day, it starts at point faces east, and walks a distance of units due east. Each night the bug rotates counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point Then where and are relatively prime positive integers. Find
Problem 9
Let be the set of positive integer divisors of Three numbers are chosen independently and at random with replacement from the set and labeled and in the order they are chosen. The probability that both divides and divides is where and are relatively prime positive integers. Find
Problem 10
Let and be positive integers satisfying the conditions
is a multiple of and
is not a multiple of
Find the least possible value of
Problem 11
For integers and let and Find the number of ordered triples of integers with absolute values not exceeding for which there is an integer such that
Problem 12
Let be the least positive integer for which is divisible by Find the number of positive integer divisors of
Problem 13
Point lies on side of so that bisects The perpendicular bisector of intersects the bisectors of and in points and respectively. Given that and the area of can be written as where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find .
Problem 14
Let be a quadratic polynomial with complex coefficients whose coefficient is Suppose the equation has four distinct solutions, Find the sum of all possible values of
Problem 15
Let be an acute triangle with circumcircle and let be the intersection of the altitudes of Suppose the tangent to the circumcircle of at intersects at points and with and The area of can be written in the form where and are positive integers, and is not divisible by the square of any prime. Find
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2019 AIME II Problems |
Followed by 2020 AIME II Problems | |
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